Knowledge Spaces, Attribute Dependencies, and Graded Knowledge States

The present paper deals with dependencies developed within the theory of knowledge spaces. Knowledge spaces represent a new paradigm in psychological approaches to assessment of knowledge. A distinguishing feature of knowledge spaces is their non-numerical character. The aim of the present paper is twofold. First, we bring up several remarks on data dependencies studied within knowledge spaces. Second, we consider the dependencies in a framework which is more general than that of classical knowledge spaces. Namely, we abandon the assumption that a knowledge state is a set of problems/questions which an individual is able to solve. Instead, we assume that a knowledge state is a graded set (fuzzy set) of problems. Our assumption accounts for situations where it is possible that an individual can solve a particular problem partially, rather than just "can solve" or "cannot solve". We propose a definition of dependencies and validity of dependencies in knowledge spaces with graded knowledge states, provide selected properties of the dependencies, and a lemma which serves as a bridge to existing results on so-called fuzzy attribute implications.